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On a Class of Distributions Stable Under Random Summation

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  04 February 2016

L. B. Klebanov ,
A. V. Kakosyan ,
S. T. Rachev  and
G. Temnov
L. B. Klebanov*
Affiliation:
Charles University
A. V. Kakosyan
Affiliation:
Yerevan State University
S. T. Rachev
Affiliation:
Stony Brook University
G. Temnov*
Affiliation:
University College Cork
*
∗ Postal address: Department of Probability and Statistics, Charles University, Prague Sokolovska 83, Prague-8, CZ 18675, Czech Republic. Email address: klebanov@chello.cz
∗∗ Postal address: School of Mathematical Sciences, University College Cork, Western Gateway Building, Western Road, Cork, Ireland. Email address: g.temnov@ucc.ie
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Abstract

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We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

Keywords

Stabilityrandom summationcharacteristic functionhyperbolic secant distribution

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 303 - 318
Copyright
© Applied Probability Trust 

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